Pub Date : 2020-10-20DOI: 10.1556/012.2020.57.3.1465
A. Alahmari, Falih Aldosray, M. Mabrouk
Let 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.
{"title":"Banach algebras satisfying certain chain conditions on closed ideals","authors":"A. Alahmari, Falih Aldosray, M. Mabrouk","doi":"10.1556/012.2020.57.3.1465","DOIUrl":"https://doi.org/10.1556/012.2020.57.3.1465","url":null,"abstract":"Let 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"90 1","pages":"290-297"},"PeriodicalIF":0.7,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84577896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-20DOI: 10.1556/012.2020.57.3.1473
Nazlı Yazıcı Gözütok, B. Ö. Güler
In this study, we investigate suborbital graphs Gu,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group (N) and then obtain some properties of the graphs arising from this action.
{"title":"Quadrilateral cell graphs of the normalizer with signature (2,4,∞)","authors":"Nazlı Yazıcı Gözütok, B. Ö. Güler","doi":"10.1556/012.2020.57.3.1473","DOIUrl":"https://doi.org/10.1556/012.2020.57.3.1473","url":null,"abstract":"In this study, we investigate suborbital graphs Gu,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group (N) and then obtain some properties of the graphs arising from this action.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"353 1","pages":"408-425"},"PeriodicalIF":0.7,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76600594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-20DOI: 10.1556/012.2020.57.3.1464
P. Kórus, L. M. Lugo, J. E. Valdés
In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions.
{"title":"Integral inequalities in a generalized context","authors":"P. Kórus, L. M. Lugo, J. E. Valdés","doi":"10.1556/012.2020.57.3.1464","DOIUrl":"https://doi.org/10.1556/012.2020.57.3.1464","url":null,"abstract":"In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"13 1","pages":"312-320"},"PeriodicalIF":0.7,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72671586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-20DOI: 10.1556/012.2020.57.3.1467
T. Ahmed
Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever A ∈ V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated HenkinOrey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that SdNrnCAω ∩ CRCAn ⊆ K ⊆ ScNrnCAn+3, where CRCAn is the class of completely representable CAns, and Sd(Sc) denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω ⊆ K ⊆ ScRaCA5 is not elementary.
定2 < n < ω，设CAn表示维数为n的圆柱代数。大致说来，CAn是一阶逻辑的证明理论的代数对立物，它被限制在前n个变量，我们用Ln表示。可表示can的RCAn的变化从代数上反映了Ln的语义。RCAn的成员是由真正的n元关系组成的具体代数，具有由关系性质引起的集合论操作，例如称为柱化的投影。尽管CAn具有有限的等式公理化，但RCAn不是有限公理化的，而且它通常表现出野性，通常是不可预测的和不守规矩的行为。这使得CAn的理论比布尔代数的理论丰富得多，就像Lω比命题逻辑丰富得多一样。我们使用所谓的膨胀和模糊结构来证明，包含和包括RCAn的几个变种(实际上是无限多的)不是原子规范的。一类带算子的布尔代数是原子正则的，如果A∈V是原子的，那么它的dedekin - macneille补全，有时称为它的最小补全，也在V中。从我们迄今为止得到的代数结果中，我们利用代数逻辑的强大机制，证明了著名的HenkinOrey省略类型定理，它是Lω，ω的模型论的经典的第一个(历史上)基石之一。对于Ln，即使我们允许某些只局部经典的广义模型，也会失败。还证明了满足SdNrnCAω∩CRCAn≥CRCAn(其中CRCAn为完全可表示的can的类，Sd(Sc)为形成致密(完全)子代数的运算)的任何类K都不是初等的。最后，我们证明了任何类K使得SdRaCAω≥≥scaca5都不是初等的。
{"title":"Atom canonicity and first order definability in classes of algebras of relations","authors":"T. Ahmed","doi":"10.1556/012.2020.57.3.1467","DOIUrl":"https://doi.org/10.1556/012.2020.57.3.1467","url":null,"abstract":"Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever A ∈ V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated HenkinOrey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that SdNrnCAω ∩ CRCAn ⊆ K ⊆ ScNrnCAn+3, where CRCAn is the class of completely representable CAns, and Sd(Sc) denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω ⊆ K ⊆ ScRaCA5 is not elementary.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"57 1","pages":"321-371"},"PeriodicalIF":0.7,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87757195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-28DOI: 10.1556/012.2021.58.1.1483
N. Gogolashvili, G. Tephnadze
In this paper we prove and discuss some new (Hp, Lp,∞) type inequalities of the maximal operators of T means with monotone coefficients with respect to Walsh–Kaczmarz system. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out. In particular, we apply these results to prove a.e. convergence of such T means.
{"title":"On the Maximal Operators of T Means with Respect to Walsh–Kaczmarz System","authors":"N. Gogolashvili, G. Tephnadze","doi":"10.1556/012.2021.58.1.1483","DOIUrl":"https://doi.org/10.1556/012.2021.58.1.1483","url":null,"abstract":"In this paper we prove and discuss some new (Hp, Lp,∞) type inequalities of the maximal operators of T means with monotone coefficients with respect to Walsh–Kaczmarz system. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out. In particular, we apply these results to prove a.e. convergence of such T means.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"45 4 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76925002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1556/012.2020.57.2.1459
G. Gát, R. Toledo
In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.
{"title":"Numerical solution of linear differential equations by Walsh polynomials approach","authors":"G. Gát, R. Toledo","doi":"10.1556/012.2020.57.2.1459","DOIUrl":"https://doi.org/10.1556/012.2020.57.2.1459","url":null,"abstract":"In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76653273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1556/012.2020.57.2.1463
Hassen Cheriha, Y. Gati, V. Kostov
Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for , where (resp.) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.
{"title":"Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs","authors":"Hassen Cheriha, Y. Gati, V. Kostov","doi":"10.1556/012.2020.57.2.1463","DOIUrl":"https://doi.org/10.1556/012.2020.57.2.1463","url":null,"abstract":"Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for , where (resp.) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"13 1","pages":"165-186"},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86602994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1556/012.2020.57.2.1454
G. Vegter, M. Wintraecken
Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.
{"title":"Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes","authors":"G. Vegter, M. Wintraecken","doi":"10.1556/012.2020.57.2.1454","DOIUrl":"https://doi.org/10.1556/012.2020.57.2.1454","url":null,"abstract":"Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85334375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1556/012.2020.57.2.1462
Wei-Feng Xuan, Yan Song
In this paper, we prove that if X is a space with a regular Gδ-diagonal and X2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that {g2(n, x): n ∈ ω} = {x} for each x ∈ X then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying Hψ(X) = κ then e(X) 22κ; and if X is Hausdorff and we(X) = Hψ(X) = κsubset of a space then e(X) 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.
{"title":"On star Lindelöf spaces","authors":"Wei-Feng Xuan, Yan Song","doi":"10.1556/012.2020.57.2.1462","DOIUrl":"https://doi.org/10.1556/012.2020.57.2.1462","url":null,"abstract":"In this paper, we prove that if X is a space with a regular Gδ-diagonal and X2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that {g2(n, x): n ∈ ω} = {x} for each x ∈ X then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying Hψ(X) = κ then e(X) 22κ; and if X is Hausdorff and we(X) = Hψ(X) = κsubset of a space then e(X) 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"329 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76566382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1556/012.2020.57.2.1460
A. Laurinčikas
Let 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.
{"title":"Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function","authors":"A. Laurinčikas","doi":"10.1556/012.2020.57.2.1460","DOIUrl":"https://doi.org/10.1556/012.2020.57.2.1460","url":null,"abstract":"Let 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"69 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82380358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：